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Special Equipment: KitchenAid(R) Stand Mixer; KitchenAid(R) Sifter + Scale Attachment; One 4-in/10-cm by 13-in/33-cm fluted tart pan with a removable bottom; 2 sheets parchment paper; rolling pin; large rimmed baking sheet; pie weights (or use dried beans or rice); small offset spatula
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Attach the KitchenAid(R) Sifter + Scale Attachment to your KitchenAid(R) mixer fitted with the flat beater. Place the scale lid on top. Weigh the 1/2 cup (4 oz) butter, cut it into small cubes, place it into a small bowl and refrigerate it while you weigh and prepare the other ingredients.
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With the ingredient valve in the closed position, weigh the flour (5.8 oz), sugar (1.75 oz) and salt (0.03 oz) in the ingredient hopper, taring the scale between ingredients. Turn the mixer to low speed, open the ingredient valve, and sift the flour mixture into the mixing bowl. While mixing, scatter the ice-cold butter over top, and mix on medium-low speed until the mixture looks pale yellow and sandy, about 2 minutes.
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In a small bowl, use a fork to beat together the cream and egg yolk. With the mixer on medium-low speed, add the cream mixture and mix to form a dough, about 45 seconds longer.
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Turn the dough out onto a sheet of parchment paper. Gather the dough and flatten it to form a thick disk. Place the second sheet of parchment over top. Using a rolling pin, roll out the dough to form a 1/4-in-/6-mm-thick long rectangle, rolling it to be a little larger than the tart pan. Place the dough, still between the parchment sheets, on a rimmed baking sheet and refrigerate until cold, about 20 minutes. Wipe out the mixer bowl and wipe clean the paddle attachment.
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While the dough is chilling, make the filling. In a small saucepan over medium-low heat, melt the 4 Tbs butter. Remove from the heat. Split the vanilla bean lengthwise and scrape out the seeds with the back of a knife. Add the seeds and pod to the melted butter and place back on the heat. Cook, swirling the pan frequently, until the butter darkens to a nutty brown and the solids drop to the bottom of the pan, about 3 to 5 minutes. Remove from the heat, discard the vanilla bean pod, and set aside to cool to room temperature, about 15 minutes.
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Stir the butter and measure out 4 tablespoons of the butter, being sure to get as much of the vanilla bean seeds as possible. Add the butter, along with the egg, to the mixer bowl fitted with the paddle attachment. (There will be a tiny amount of browned butter leftover; don't waste it-smear it on toast!)
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With the ingredient valve in the closed position, weigh the sugar (2.2 oz), flour (1 oz), and salt (0.05 oz) in the ingredient hopper. Turn the mixer to medium-low speed and beat the butter and egg until combined, about 1 minute. Open the ingredient valve, and sift in the flour mixture. Mix just until the flour disappears. Remove the mixer bowl, and use a rubber spatula to blend in any flour remaining on the sides of the bowl. Transfer the filling to a small bowl and set aside at room temperature.
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Position an oven rack in the center of the oven and preheat the oven to 400u00b0F/200u00b0C/gas 6.
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Remove the dough from the refrigerator. Lift off the top sheet of parchment and set it aside. Flip the tart dough over onto the tart pan, using the bottom parchment to ease it into the pan, gently pressing the dough into the bottom, corners, and up the sides of the pan. Gently lift off the parchment. Use your fingers to patch any cracks in the dough. Roll the rolling pin across the top of the tart pan to cut off excess dough. The dough should be flush with the top of the pan. Using a fork, prick the bottom of the tart all over. Using the top sheet of parchment, fit it into the shell. Freeze the tart shell for 20 minutes.
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Prebake the tart shell, lined with parchment and filled with pie weights, dried beans, or rice, until the shell looks set and lightly golden, about 12 minutes. Carefully, lift out the parchment and pie weights. Reduce the oven temperature to 350u00b0F/180u00b0C/gas 4. Continue baking the tart shell until the bottom is set and lightly golden, about 10 minutes longer. Transfer the tart shell to a wire rack to cool for 10 minutes before adding the filling.
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Spoon the filling into pre baked tart shell, spreading it evenly over the bottom with a small offset spatula or spoon. Arrange the berries in even rows on top. Bake on the center rack until the filling puffs up around the fruit and is golden brown in the center as well as along the edges, about 35 to 40 minutes.
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Transfer to a wire rack to cool. Cut the tart crosswise to serve. Dust each serving with confectioners' sugar. The tart is best when served on the day it is baked. (The tart shell can be frozen, then well wrapped and frozen, for up to 1 month. Bake it, straight from the freezer, as instructed above, on the day you plan to serve it. The filling can be made 1 day in advance. Cover and refrigerate. Remove from the refrigerator and bring to room temperature before filling the tart.
